Hwang, Jin-Soo

Nakagiri, Shin-ichi

Kyoto University Research Information Repository

TitleBoundary control problems for viscoelastic systems with longmemory (Mathematical models and dynamics of functionalequations)Author(s)Hwang, Jin-Soo; Nakagiri, Shin-ichiCitation数理解析研究所講究録 (2004), 1372: 183-189Issue Date2004-04URL http://hdl.handle.net/2433/25511RightType Departmental Bulletin PaperTextversionpublisherKyoto University183Boundary control problems for viscoelasticsystems with long memory$\mathrm{J}\mathrm{j}\mathrm{n}$-Soo Hwang ( )Division of Mathematical and Material Science,The Graduate School of Science and Technology, Kobe UniversityShin-ichi Nakagiri ( )Department of Applied Mathematics, Faculty of Engineering, Kobe University, JAPAN.1 IntroductionIn the study of viscoelastic materials, the state of stresses at the instant $t$ defends on thestrain at the instant $t$ , but also on the strains at the instants previous to the present instant$t$ . $\mathrm{b}\mathrm{o}\mathrm{m}$ this standpoint of view, the viscoelastic equations with long memory are introduced.The qualification of long memory is given by the Volterra integrals on the effects of memory ofmaterials. We shall give the description of the linear viscoelastic systems with long memory inthe three dimensional Euclidean space $\mathrm{R}^{3}$ .Let $\Omega$ be an open and bounded set in $\mathrm{R}^{3}$ with sufficiently smooth boundary $\partial\Omega$ . Let $T>$$0$ be fixed and let $Q=$ $(0, T\mathrm{I})$ $\cross\Omega$ and $\Sigma=(0,T)\cross$ $\partial\Omega$ . We denote by $y=(y_{1}, y_{2},y_{3})$ $\mathrm{a}$displacement field in $\mathrm{R}^{3}$ and Oijkh are the coefficients of instantaneous elasticity. The system oflinear viscoelastic equations with long memory is described by$\frac{\partial^{2}y_{i}}{\partial t2}-\frac{\partial}{\partial x_{j}}a$ijkh$\epsilon_{kh}(y)-\int_{0}^{t}\frac{\partial}{\partial x_{j}}b_{ijkh}(t-s,x)\epsilon_{kh}(y)ds=fi,$ $i=1,2,3$ , (1. 1)where $\epsilon lkh(y)=\frac{1}{2}(\frac{\partial y_{h}}{\partial x_{k}}+\frac{\partial y_{k}}{\partial x_{h}})$ is a strain tensor element, $b_{ijkh}$ are the coefficients of elasticity bytaking into the memory effects of the material, $f=(f1, f_{2}, f_{3})$ is an external force. Throughoutthis paper we assume that the coefficients aijkh and $b_{ijkh}$ satisfy$\{\begin{array}{l}a_{\dot{\iota}jkh},b_{ijkh}\in L^{\infty}(Q)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{u}i,j,k,ha_{jkh}=a_{j\dot{l}kh}=a_{kh\dot{l}j}a_{\dot{l}jkh}\xi_{jj}\xi_{kh}\geq\alpha\xi_{ij}\xi_{ij},\exists\alpha>0,\forall\xi_{ij}\in \mathrm{R},\xi_{ij}=\xi_{j}..\cdot\end{array}$ (1.2)and that $a_{ijkh}$ and $b_{\dot{*}jkh}$ have the following time regularities:$\{$$tarrow$ Oijkh $(t$ , $\cdot$ $)$ $\in L^{\infty}(\Omega)$ is continuousely differentiable and$\frac{\partial a_{\dot{l}jkh}}{\partial t}\in L^{\infty}(Q)$ for all $i,j$ , $k$ , $h$ ;$tarrow b_{\dot{l}jkh}(t$ , $\cdot$ $)$ $\in L^{\infty}(\Omega)$ is continuousely differentiable and$\frac{\partial b_{\dot{l}jkh}}{\partial t}\in L^{\infty}(Q)$ for au $i,j$ , $k$ , $h$ .(1.3)1372 2004 183-189184The purpose of this paper is to study the boundary control problems for the viscoelasticsystem (1.1). First we consider the following Neumann boundary control system$\{$$\frac{\partial^{2}y_{i}(v)}{\partial t^{2}}-\frac{\partial}{\partial x_{j}}aijkh^{f}kh(\mathrm{J}(\mathrm{v})-\int_{0}^{t}\frac{\partial}{\partial x_{j}}b_{ijkh}(t-s, x)\epsilon_{kh}$( 1 (v)) $)ds=f_{i}$ in $Q$ ,$[a_{ijkh} \epsilon_{kh}(y(v))+\int_{0}^{t}b_{ijkh}(t-s, x)\mathrm{c}_{kh}(y(\mathrm{f}>))(\# s]\mathrm{n}j= /)i$ on Et,$1\mathrm{i}(v;0, x)=yi$ , $\frac{\partial y_{i}}{\partial t}(v;0, x)=l_{j}$ in $\Omega$ , $i=1,2,3$,(1.4)where $f=(f_{1}, f_{2},f_{3})\in[L^{2}(0,T;(H^{1}(\Omega))’)]^{3}$, $y_{0}=(y_{1}^{0}, y_{2}^{0}, y_{3}^{0})\in[L^{2}(\Omega)]^{3}$ , $y_{1}$ $=(y_{1}^{1},y_{2}^{1},y_{3}^{1})\in$$[(H^{1}(\Omega))’]^{3}$ , ni is the $j$-th outward normal to $\partial\Omega$ , and the boundary control variables $v=$$(v_{1},v_{2},v\mathrm{s})$ are assumed to satisfy the condition$v:\in L^{2}(\Sigma)$ , $i=1,2,3$ . (1.5)It is verified by the method of transposition (cf. Lions [2], Lions and Magenes [3], that there isa unique transposed solution $y(v)\in[L^{2}(Q)]^{3}$ of (1.4) for each $v$ satisfying (1.5). Therefore, forthe controlled system (1.4) we can attach the following quadratic cost functional given by$J(v)= \sum_{i=1}^{3}\int_{Q}(yi(v)-zdi)^{2}dxdt+\sum_{i=1}^{3}\nu i\int_{\Sigma}vi^{2}$dxdt, (1. 6)where $zd$: are desired values in $L^{2}(Q)$ and $\nu i>0,i=1,2,3$ .Next we consider the following Dirichlet boundary control system$\{$$\frac{\partial^{2}y_{i}(v)}{\partial t^{2}}-\frac{\partial}{\partial x_{j}}a_{ij}tkh^{C}tkh(y(v))-\int_{0}^{t}\frac{\partial}{\partial x_{j}}b_{\dot{l}jkh}(t-s,x)\epsilon_{kh}(y(v))ds=f_{i}$ in $Q$ ,$y_{i}(v)=v:$ on $\Sigma$ ,$li(v;0, x)=/i$ , $\frac{\partial y_{i}}{\partial t}(v;0,x)--y_{i}^{1}$ in $\Omega$ , $i=1,2,3$ ,(1.7)where $f=(f1, f_{2}, f_{3})\in[L^{2}(Q)]^{3}$ , $y_{0}=(y_{1}^{0}, y_{2}^{0},y_{3}^{0})\in[H^{1}(\Omega)]^{3}$ and $y_{1}=(y_{1}^{1},y_{2}^{1}, y_{3}^{1})\in[L^{2}(\Omega)]^{3}$ .Further in (1.7) we assume the stronger regularity condition on $v=(v_{1}, v2,v_{3})$ such that$v_{i}\mathrm{E}$ $H_{0}^{2}(\Sigma)$ , $i=1,2,3$ . (1.8)We can verify that there exists a unique weak solution $y(v)$ of (1.7) in the sense of Dautrayand Lions [1] for each $v$ satisfying (1.8). The solution $y(v)$ has the regularity $y(v)\in[L^{2}(Q)]^{3}$ ,$y(v)\in[C([0,T];L^{2}(Q))]^{3}$ . Hence, for the control system (1.7) we can attach the following twotypes of quadratic cost functional:$J(v)$ $=$ $\sum_{\dot{v}=1}^{3}\int_{Q}(y_{i}(v)-z_{di})^{2}dxdt+.\sum_{|=1}^{3}\nu_{i}||v_{\dot{l}}||_{H_{0}^{2}(\Sigma)}^{2}$; (1.9)$J(v)$ $=$ $\sum_{\dot{l}=1}^{3}\int_{\Omega}(y\mathrm{i}(v;1 )-zd:)^{2}dx+\sum_{\dot{\iota}=1}^{3}\nu_{i}||\mathrm{t})\mathrm{i}||\mathrm{L}_{3}(\Sigma)$ . (1.10)In (1.9) and (1.10) $z_{d:}$ are desired values in $L^{2}(Q)$ and $L^{2}(\Omega)$ , respectively, and $\nu_{\dot{l}}>0,i=1,2,3$ .In this paper we establish the necessary conditions of optimality both for the Neumannboundary control system (1.4) with the cost (1.6) and the Dirichlet boundary control system(1.7) with the cost (1.9) or (1.10) by introducing proper adjoint systems.1852 Neumann boundary control problemsIn this section we study the Neumann boundary control problems of (1.4). To formulate theproblems, we need to introduce the transposed solution of (1.4) by the transposition method.Lem a 2.1 Assume that $f\mathrm{E}$ $[L^{2}(0, T;(H^{1}(\Omega))’)]^{3}$ , $y0\in[L^{2}(\Omega)]^{3}$ , $y_{1}\in[(H^{1}(\Omega))’]^{3}$ and (1.5)in the system (1.4). Then there exists a unique element $y$ $=(y_{1}, y_{2}, ys)$ in $[L^{2}(Q)]^{3}$ such that$\{$$\sum_{i=1}^{3}\int_{Q}y_{\dot{\mathrm{t}}}(\frac{\partial^{2}\phi_{i}}{\partial t^{2}}-\frac{\partial}{\partial x_{h}}a_{\mathrm{i}jkh}\epsilon_{\mathrm{i};\mathrm{i}}(\phi)-\int_{t}^{T}\frac{\partial}{\partial x_{h}}b_{ijkh}(s-t, x)\epsilon_{ij}(\phi)ds)dxdt$$= \sum_{i=1}^{3}\int_{Q}f_{i}\phi_{i}dxdt-\sum_{i=1}^{3}\int_{\Omega}y_{i}^{0}\frac{\partial\phi_{i}}{\partial t}(0, x)dx+\sum_{i=1}^{3}\int_{\Omega}y_{l}^{1}\phi_{i}(0, x)dx+\sum_{i=1}^{3}\int_{\Sigma}v_{i}\phi_{i}d\Sigma$for all function $\phi$ such that $in X,$ where$X=$ $\{$ $\phi=(\phi_{1}, \phi_{2}, \phi_{3})|\phi_{i}\in L^{2}(0,T;H^{1}(\Omega))$ ,$\frac{\partial^{2}\phi_{\dot{l}}}{\partial t^{2}}-\frac{\partial}{\partial x_{h}}a_{ijkh}\epsilon_{\dot{\iota}j}(\phi)-\int_{t}^{T}\frac{\partial}{\partial x_{h}}b_{ijkh}(s-t, x)\epsilon_{\dot{l}j}(\phi)ds\in L^{2}(Q)$ ,$[a_{ijkh} \epsilon_{ij}(\phi)+\int_{t}^{T}b_{jjkh}(s-t, x)\epsilon_{ij}(\phi)ds]\mathrm{n}_{h}=0$ on $\Sigma$ ,$\phi_{i}(T, x)=\frac{\partial\phi}{\partial}i(T, x)=0,$ $i=1,2,3\}$ .Here we note that$\phi_{i}\in C([0,T];H^{1}(\Omega))$ , $\frac{\partial\phi_{i}}{\partial t}\in C([0, T];L^{2}(\Omega))$ , $\phi_{i}|\Sigma\in H^{\frac{1}{2}}(\Sigma)\subset L^{2}(’)$for all $\phi=(\phi_{1}, \phi_{2}, \phi_{3})\in X.$By Lemma 2.1, for the system (1.4) we can consider the cost given by$J(v)= \sum_{\dot{\iota}=1}^{3}\int_{Q}(y_{i}(v)-z_{di})^{2}dxdt+\sum_{i=1}^{3}\nu_{\dot{l}}\int_{\Sigma}v_{i}^{2}$ d\Sigma , $v\in[L^{2}(\Sigma)]^{3}$ , (2.1)where $\nu_{i}>0$ and $z_{di}\in L^{2}(Q)$ , $i=1,2,3$ . Let $\mathcal{U}_{ad}\subset[L^{2}(\Sigma)]^{3}$ be a closed and convex set ofadmissible controls. The element $u\in \mathcal{U}_{ad}$ such that$v\in \mathcal{U}_{ad}\mathrm{i}dJ(v)=J(u)$(2.2)is called the optimal control. It is easily verified that the optimal control $u$ for the cost (2.1)exists uniquely by the positivity $\nu_{i}>0$ for $i=1,2,3$. Then the optimality condition is given by$\sum_{i=1}^{3}\int_{Q}(y_{i}(u)-z_{d\mathrm{i}})(y_{i}(v)-y_{i}(u))dxdt+\sum_{=j1}^{3}\nu_{i}\int_{Q}(u_{i})(v:-u_{i})d\Sigma\geq 0,$ $lv\in I_{ad}$ , (2.3)where tt is the optimal control for (2.1). We want to write down the condition (2.1) in terms ofadjoint state equation. For this, we introduce the adjoint system by$\{$$\frac{\partial^{2}p(u)}{\partial 2}i-\frac{\partial}{\partial x_{h}}a_{ijkh^{E}ij}$ $?(u))- \int_{t}^{T}\frac{\partial}{\partial x_{h}}b_{ijkh}(s-t, x)\epsilon_{\dot{\iota}j}(p(u))ds=y:(u)-z_{d}$ in $Q$ ,$[a_{ijkh} \epsilon_{\dot{l}j}(p(u))+\int_{t}^{T}b_{ijkh}(s-t,x)\epsilon_{ij}(y(u))ds]\mathrm{n}_{h}=0$ on $\Sigma$ , (2.4)$p_{i}(u;T, x)=0,$ $\frac{\partial p}{\partial}i(u;T, x)=0$ in $\Omega$ , $i=1,2,3$ .188Since $y_{i}(u)-zdi\in L^{2}(Q)$ , $i=1,2,3$ by assumption, we can verify that the weak solution$p=$ ($p_{1}$ , $p_{2}$ , ps)\in X of (2.4) exists uniquely. The optimality condition can be obtained by thefollowing theorem.Theorem 2.1 The optimal control $u\in Ia’ d$ $\subset[L^{2}(\Sigma)]^{3}$ for the cost (2.1) is characterized by thefollowing system of equations and inequality:$\{$$\frac{\partial^{2}y_{i}(v)}{\partial t^{2}}-\frac{\partial}{\partial x_{j}}a$i$ikh^{f}kh(y(v))- \int_{0}^{t}\frac{\partial}{\partial x_{j}}b_{ijkh}(t-s, x)\epsilon_{kh}(y(v))ds=f_{i}$ in $Q$ ,$[a_{ijkh}e_{kh}(l(v))$ $+ \int_{0}^{t}b_{ijkh}(t-s, x)\epsilon_{kh}(y(v))ds]\mathrm{n}j=v_{i}$ on $\Sigma$ ,$yi(v;0, x)= \oint_{l}(x)$ , $\frac{\partial y_{i}}{\partial t}(v;0, x)=y^{1}i(x)$ in $\Omega$ , $i=1,2,3$,$\{$$\frac{\partial^{2}p\cdot(u)}{\partial 2}i-\frac{\partial}{\partial x_{h}}a_{ijkh}\epsilon_{ij}(p(u))-\int_{t}^{T}\frac{\partial}{\partial x_{h}}b_{ijkh}(s -t, x)\epsilon_{ij}(p(u))ds=y_{j}(u)-z_{di}$ in $Q$ ,$[a_{ijkh} \epsilon_{ij}(p(u))+\int_{t}^{\tau_{b_{ijkh}(s-t,X)\epsilon}}ij(p(u))ds]\mathrm{n}h=0$ on $\mathrm{c}$ ,$p_{i}(u;T, x)=0,$ $\frac{\partial p_{i}}{\partial t}(u;T, x)=0$ in $\Omega$ , $i=1,2,3$,$\sum_{i=1}^{3}\int_{\Sigma}(p_{i}(u)+\nu iui)$ $(vi-u_{i})d\Sigma\geq 0,$ $\forall v=(v1, v2, v\mathrm{s})$ $\in \mathcal{U}_{ad}\subset[L^{2}(\Sigma)]^{3}$ .Example 2.1 Assume that the admissible set $u_{ad}$. is given by$\mathcal{U}_{ad}=\{v=(v_{1},v2, v_{3})|vi\geq 0$ on $\Sigma$ , $i=1,2,3\}$ .Then by Theorem 2.1 the optimal control/ $=$ $(u_{1}, u_{2},23)$ is given by$u_{i}=’ a_{ijkh} \epsilon_{kh}(y)+\int_{0}^{t}b_{ijkh}(t-s, x)\epsilon kh(y)ds]\mathrm{n}j$, $i=1,2,3$ ,where !/ is the solution of the following unilateral problem on $y$ and $p$ :$\{$$\frac{\partial^{2}y_{i}}{\partial t^{2}}-\frac{\partial}{\partial x_{j}}a_{ijkh}\epsilon_{kh}(y)-\int_{0}^{t}\frac{\partial}{\partial x_{j}}b_{ijkh}(t-s, x)\epsilon_{kh}(y)ds=$ $\mathrm{f}_{\mathrm{i}}$ in $Q$ ,$\frac{\partial^{2}p_{i}}{\partial t^{2}}-\frac{\partial}{\partial x_{h}}a_{ijkh}\epsilon ij(p)-\int_{t}^{T}\frac{\partial}{\partial x_{h}}b_{ijkh}(s-t,x)\epsilon ij(p)ds=y_{i}-z_{di}$ in $Q$ , $i=1,2,3$ ,$\{$$[a_{ijkh}c_{tkh}(y)+ \int_{0}^{t}b_{ijkh}(t-s,x)\epsilon kh(y)ds]\mathrm{n}j\geq 0$ on $\Sigma$ ,$p_{i}+ \nu_{i}[a_{ijkh}\epsilon_{kh}(y)+\int_{0}^{t}b_{ijkh}(t-s,x)\epsilon kh(y)ds]\mathrm{n}j\geq 0$ on $\Sigma$ ,$[a_{ijkh}e_{ij}p)$ $+ \int_{t}^{\tau_{b_{ijkh}(s-t,x)\epsilon}}ij(p)d_{S}]\mathrm{n}h=$ o on $\mathrm{C}$ ,$[a_{ijkh} \epsilon_{kh}(y)+\int_{0}^{t}b_{\dot{0}jkh}(t-s,x)\epsilon_{kh}(y)ds]\mathrm{n}j\mathrm{x}$($p_{\dot{l}}+\nu_{\dot{l}}[a_{ijkh}\epsilon_{kh}(j)$ $+ \int_{0}^{t}$ bijkh $(t-s, x)\epsilon kh(y)ds]\mathrm{n}j)=0,$ on $\Sigma$ , $i=1,2,3$ .$\{$$y\mathrm{n}(0, x)=$ !/?(x), $\frac{\partial y_{i}}{\partial t}(0, x)=y^{1},(x)$ in $\Omega$ ,$p_{i}(T,x)=0,$ $\frac{\partial p_{i}}{\partial t}(T,x)=0$ in $\Omega$ , $i=1,2,3$ .We note that the above equations md inequalities are given in the sense of distribution.1873 Dirichlet boundary control problemsIn this section we consider the Dirichlet boundary control system (1.7). At first, we define aninner product of $[H_{0}^{2}(\Sigma)]^{3}$ by$( \phi, 7)_{[H_{0}^{2}(\mathrm{I})]^{3}}=\sum_{i=1}^{3}\int_{\Sigma}\triangle \mathrm{r}\phi_{i}(t, x)\triangle \mathrm{r}\psi_{i}(t, x)d\Gamma dt+\sum_{i=1}^{3}\int_{\Sigma}\frac{\partial^{2}}{\partial t^{2}}\phi_{i}(t, x)\frac{\partial^{2}}{\partial t^{2}}\psi_{i}(t, x)$d\Gamma dt,where $\triangle \mathrm{r}$ is the Laplace-Beltrami operator on $\Gamma=$ an.For each $v$ $=(v_{1}, v_{2}, v_{3})$ satisfying (1.8) we can construct $\varphi=(\varphi_{1}, \mathrm{p}_{2}, \varphi_{3})$ such that$\{$$\varphi_{i}\in H^{2}(\overline{Q})$ , $\varphi_{i}=v_{i}$ on $\Sigma$ ,$\varphi i(0,x)=\frac{\partial}{\partial t}\varphi i(0, x)=0,$ $i=1,2,3$ .Let $zi=yi(v)-\varphi i.$ Then we have the homogeneous Dirichlet boundary problem$\{$$\frac{\partial^{2}z_{i}(v)}{\partial t^{2}}-\frac{\partial}{\partial x_{j}}a_{\dot{l}jkh}\epsilon_{kh}(z(v))-\int_{0}^{t}\frac{\partial}{\partial x_{j}}b_{ijkh}(t-s, x)\epsilon_{kh}(z(v))ds=g_{i}$ in $Q$ ,$z_{i}(v)=0$ on $\Sigma$ ,$z_{i}(v;0, x)=y_{l}^{0}(x)$ , $\frac{\partial z_{i}}{\partial t}(v;0, x)=y_{i}^{1}(x)$ in $\mathrm{Q}$ , $i=1,2,3$ ,(3.1)where$g_{i}=f_{\mathrm{i}}-$ [ $\frac{\partial^{2}\varphi_{i}(v)}{\partial t^{2}}-\frac{\partial}{\partial x_{j}}a$i$jkh \epsilon_{kh}(\varphi(v))-\int_{0}t$ $\frac{\partial}{\partial x_{j}}b_{ej}tkh(t-s,x)\epsilon_{kh}(\varphi(v))ds$] $\in L^{2}(Q)$ , $i=1,2,3$ .The system (3.1) admit a unique weak solution $z=(z_{1}(v), z_{2}(v),$ $z_{3}(v))$ under the conditions$f\in[L^{2}(Q)]^{3}$ , $y0\in[H^{1}(\Omega)]^{3}$ and $y1\in[L^{2}(\Omega)]^{3}$ and (1.8) (cf. Dautray and Lions [1]). Thus wehave the solutions $y_{/}$. $=z_{i}(v)1?\mathrm{i}$ , $i=1,2,3$ of (1.7). Hence $y\in[L^{2}(Q)]^{3}$ and $y(T)\in[L^{2}(\Omega)]^{3}$follow.3.1 Case of distributive value observationsIn this case the cost functional is given by$J(v)= \sum_{i=1}^{3}\int_{Q}(y_{i}(v)-z_{di})^{2}dxdt+\sum_{i=1}^{3}\nu_{i}||v_{i}||_{H_{0}^{2}(\Sigma)}^{2}$ , $\nu_{i}>0$ , $i=1,2,3$ , (3.2)where $zdi\in L^{2}(Q)$ , $i=1,2,3$ . Let $Uad$ be a closed and convex subset of $H_{0}^{2}(\Sigma)$ . Then thereexists a unique optimal control $u\in \mathcal{U}_{ad}$ for the cost (3.2). The optimal control $u=(u_{1}, u_{2},u_{3})$is characterized by$\sum_{\dot{\iota}=1}^{3}\int_{Q}(yi(u)-zdi)(y_{\dot{l}}(v)-yi(u))dxdt+\sum_{i=1}^{3}\nu_{/}.(uj, v_{\dot{l}}-u_{\dot{|}})_{H_{0}^{2}(\Sigma)}\geq 0,$ $\forall v=(v_{1}, v_{2},v\mathrm{s})\in \mathcal{U}_{ad}$ .We introduce the adjoint system by$\{$$\frac{\partial^{2}p_{i}(u)}{\partial t^{2}}-\frac{\partial}{\partial x_{h}}a_{\dot{l}jkh}\epsilon_{ij}(p(u))-\int_{t}^{T}\frac{\partial}{\partial x_{h}}b_{\dot{*}jkh}(s-t,x)\epsilon_{\dot{\iota}j}(p(u))ds=y_{\dot{\iota}}(u)-z_{d\dot{*}}$ in $Q$ ,$p:(u)=0$ on $\Sigma$ ,$p:(u;T,x)=0,$ $\frac{\partial p}{\partial}i.(v;T, x)=0$ in $\Omega$ , $i=1,2,3$ ,188where $yi(u)-zdi\in L^{2}(Q)$ , $i=1,2,3$ . There exists a unique weak solution $p=p(u)$ of thisadjoint system. Hence we have the following optimality condition for the cost (3.2).Theorem 3.1 The optimal control $u\in$ $l_{ad}$ $\subset[H_{0}^{2}(\Sigma)]^{3}$ or the cost (3.2) is characterized by thefollowing system of equations and inequality:$\{$$\frac{\partial^{2}y_{i}(u)}{\partial t^{2}}-\frac{\partial}{\partial x_{j}}a_{ijkh}\epsilon_{kh}(y(u))-\int_{0}^{t}\frac{\partial}{\partial x_{j}}b_{ijkh}(t-s, x)\epsilon_{kh}(y(u))ds=f_{i}$ in $Q$ ,$y_{i}(u)=U_{i}^{t}$ on $\Sigma$ ,$yi(u;0, x)=y0\iota’$ $\frac{\partial y_{i}}{\partial t}(u;0,x)=y\#$ in $\Omega$ , $i=1,2,3$ ,$\{$$\frac{\partial^{2}p_{i}(u)}{\partial t^{2}}-\frac{\partial}{\partial x_{h}}a_{jjkh}\epsilon_{\dot{l}j}(p(u))-\int_{t}^{T}\frac{\partial}{\partial x_{h}}b_{ijkh}(s-t,x)\epsilon_{ij}(p(u))ds=y.\cdot(u)-z_{di}$ in $Q$ ,$p_{i}(u)=0$ on $\Sigma$ ,$pi(u;T,x)=0,$ $\frac{\partial p_{\dot{f}}}{\partial t}(\mathrm{t} ;7 , x)=0$ in $\Omega$ , $i=1,2,3$ ,$\sum_{i=1}^{3}\int_{\Sigma}[-a_{\dot{l}jkh}\epsilon_{ij}(p)-\int_{t}^{T}b_{ijkh}(s-t,x)\epsilon_{\dot{l}j}(p)ds]\mathrm{n}_{h}(v_{\dot{l}}-u_{i})d\Sigma$$+ \sum_{i=1}^{3}$ $/ \Sigma\nu_{i}((\Delta_{\Gamma}+\frac{\partial^{2}}{\partial t^{2}})u_{i})((\Delta_{\Gamma}+\frac{\partial^{2}}{\partial t^{2}})(v_{i}-u_{i}))d\Sigma\geq 0,$ $\forall v\in \mathcal{U}_{ad}$ .3.2 Case of terminal value observationsIn this case the cost functional is given by$J(v)= \sum_{\dot{l}=1}^{3}\int_{\Omega}(y_{i}(v;T)-z_{di})^{2}dx+\sum_{i=1}^{3}\nu_{\mathrm{j}}||$ $\mathrm{t}_{\mathrm{j}}||\mathrm{K}_{2(\Sigma)}$ , $\forall v=(v_{1},v_{2},v_{3})\in[H_{0}^{2}(\Sigma)]^{3}$ , (3.3)where $z_{di}\in L^{2}(\Omega)$ , $\nu i>0$ , $i=1,2,3$ . Let $\mathcal{U}_{ad}$ be a closed and convex subset of $H_{0}^{2}(\Sigma)$ . Thenthe optimal control $u=$ $(u_{1,2}L,ua)$ for the cost (3.3) exists uniquely and is characterized by$\sum_{i=1}^{3}\int_{\Omega}(y_{\dot{l}}(u;T)-z_{di})(y_{i}(v;\mathrm{i}) -y_{i}(u;\#)\mathrm{E}x+\sum_{\dot{l}=1}^{3}\nu_{i}(u_{ii}, l)-u_{\dot{l}})_{H_{0}^{2}(\Sigma)}\geq 0,$ $\mathit{7}\tau)=(v_{1},12, v_{3})\in \mathcal{U}_{ad}$ .We introduce the adjoint system by$\{$$\frac{\partial^{2}p_{i}(u)}{\partial t^{2}}-\frac{\partial}{\partial x_{h}}aijkh\mathrm{e}\mathrm{i}j(p(u))-$ $7^{T}$ $\frac{\partial}{\partial x_{h}}b,jlkh(s -t,x)\epsilon ij(p(u))ds=0$ in $Q$ ,$p_{i}(u)=0$ on $\Sigma$ ,$pi(u;T,x)=0$ in $\Omega$ ,$\frac{\partial p}{\partial}i(u;T, x)=y:(u;T, x)-z_{di}(x)$ in $\Omega$ , $i=1_{1}2,3$ .(3.4)Since $y_{\dot{l}}(u;T)$ $-zd\dot{l}\in L^{2}(\Omega)$ , $i=1,2,3$ by assumption, we can obtain the unique weak solution$p=p(u)$ of (3.4). Hence the optimality condition can be obtained by the following theorem.1 $8\theta$Theorem 3.2 The optimal control $u\in \mathcal{U}_{ad}\subset[H_{0}^{2}(\Sigma)]^{3}$ for the cost (3.3) is characterized bythe following system of equations and inequality:$\{$$\frac{\partial^{2}y_{i}(u)}{\partial t^{2}}-\frac{\partial}{\partial x_{j}}a_{ijkh}\epsilon_{kh}(y(u))-\int_{0}^{t}\frac{\partial}{\partial x_{j}}b_{ijkh}(t-s, x)\epsilon_{kh}(y(u))ds=f_{i}$ in $Q$ ,$y_{i}(u)=LL_{i}$ on $\Sigma$ ,$y\mathrm{i}(u;0, x)=y$rf(x), $\frac{\partial y_{i}}{\partial t}(u;0,x)=y_{i}^{1}(x)$ in $\Omega$ , $i=1,2,3$ ,$\{$$. \frac{\partial^{l}p_{i}(u)}{\partial t^{2}}-\frac{\partial}{\partial x_{h}}a_{ijkh}\epsilon_{\dot{l}j}(p(u))-\int_{t}^{T}\frac{\partial}{\partial x_{h}}$$y_{jkh}(s-t, x)\epsilon_{\dot{l}j}(p(u))ds=0$ in $Q$ ,$p_{i}(u)=0$ on $\Sigma$ ,$p_{i}(u;T, x)=0$ in $\Omega$ ,$\frac{\partial p}{\partial}i.(u;T, x)=y\mathrm{i}(u;T, x)-zdi(x)$ in 0, $i=1,2,3$,$\sum_{i=1}^{3}\int_{\Sigma}[a_{ijkh}\epsilon_{ij}(p(u))+\int_{t}^{\tau_{b_{ijkh}(s-t,x)\epsilon_{ij}(p(u))d_{S}]\mathrm{n}}}h(vi-u_{i})d\Sigma$$+ \sum_{i=1}^{3}\int_{\Sigma}\nu_{i}((\Delta_{\Gamma}+\frac{\partial^{2}}{\partial t^{2}})u_{i})((\Delta_{\Gamma}+\frac{\partial^{2}}{\partial t^{2}})(v_{i}-u_{i}))d\Sigma\geq 0$ , $\forall v\in \mathcal{U}_{ad}\subset[H_{0}^{2}(\Sigma)]^{3}$ .References[1] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Scienceand Technology, Springer-Verlag, Vol. 5, Evolution Problems, 1992.[2] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,Springer-Verlag Berlin Heidelberg Newt York, 1971.[3] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and ApplicationsI, II, Springer-Verlag Berlin Heidelberg New York, 1972.

Boundary control problems for viscoelastic systems with long memory (Mathematical models and dynamics of functional equations)

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Boundary control problems for viscoelastic systems with long memory (Mathematical models and dynamics of functional equations)

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